Analysis of Error Metrics of Different Levels of Compression on Modified Haar Wavelet Transform

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					                                                          (IJCSIS) International Journal of Computer Science and Information Security,
                                                          Vol. 9, No. 1, January 2011
                                                                                                                                              1




        Analysis of Error Metrics of Different Levels of
           Compression on Modified Haar Wavelet
                          Transform
                   1
                       T.Arumuga Maria Devi,Assistant Professor,Centre for Information Technology and
                          Engineering,Manonmaniam Sundaranar University,Tirunelveli .TamilNadu.

            2
              S.S.Vinsley, Student IEEE Member,Guest Lecturer,Manonamaniam Sundaranar University,Centre
        for Information Technology and Engg,Manonmaniam Sundaranar University, Tirunelveli.TamilNadu.


                                                                            contains a large amount of spatial redundancy in plain areas
                                                                            where adjacent picture elements (pixels, pels) have almost
   Abstract—Lossy compression techniques are more efficient in              same values. It means that the pixel values are highly
terms of storage and transmission needs. In the case of Lossy               correlated [31]. The basic measure for the performance of a
compression, image characteristics are usually preserved in the
                                                                            compression algorithm is Compression Ratio (CR). In a lossy
coefficients of the domain space in which the original image is
transformed. For transforming the original image, a simple but              compression scheme, the image compression algorithm
efficient wavelet transform used for image compression is called            should achieve a tradeoff between compression ratio and
Haar Wavelet Transform. The goal of this paper is to achieve                image quality [32]. The balance of the paper is organized as
high compression ratio in images using 2 D Haar Wavelet                     follows: In section II describes the properties and advantages
Trnasform by applying different compression thresholds for the              of haar wavelet transformation. In section III considers the
wavelet coefficients and these results are obtained in fraction of
seconds and thus to improve the quality of the reconstructed                procedure for haar wavelet transformation. In section IV,
image ie., to arrive at an approximation of our original image.             Implementation methodologies for wavelet compression and
Another approach for lossy compression is, instead of                       linear algebra are discussed. In section V describes the
transforming the whole image, to separately apply the same                  algorithm for its implementation. In section VI considers the
transformation to the regions of interest in which the image                comparison for metrics. In section VII considers the graphs,
could be devided according to a predetermined characteristic.
                                                                            HWT image compression output results, and these results are
The Objective of the paper deals to get the coefficients is nearly
closer to zero. More specifically, the aim of the thesis is to              plotted between various parameters.           In section VIII
exploit the correlation characteristics of the wavelet coefficients         elaborates on the importance of this paper, some applications
as well as second order characteristics of images in the design of          and its extensions. Quality and compression can also vary
improved lossy compression systems for medical images. Here a               according to input image characteristics and content. Images
modified simple but efficient calculation schema for Haar                   need not be reproduced exactly. An approximation of the
Wavelet Transform.
                                                                            original image is enough for most purposes, as long as the
  Index Terms—Haar Wavelet Transform – Linear Algebra                       error between the original and the compressed image is
Technique – Lossy Compression Technique – MRI                               tolerable. Lossy compression technique can be used in this
                                                                            area.

                         I.   INTRODUCTION
                                                                                  II. PROPERTIES AND ADVANTAGES OF HAAR WAVELET
                                                                                                    TRANSFORM
I   Nrecent years, many studies have been made on wavelets.
   An excellent overview of what wavelet have brought to the                The Properties of the Haar Transform are described as
                                                                            follows:
fields as diverse as biomedical applications, wireless
communication, computer graphics or turbulence [30]. Image
compression is one of the most visible applications of                            Haar Transform is real and orthogonal. Therefore
wavelets. The rapid increase in the range and use of                               Hr=Hr*                                           (1)
electronic imaging justifies attention of systematic design of                     Hr-1 = Hr T                                       (2)
an image compression system and for providing the image                            Haar Transform is a very fast transform .
quality needed in different applications. A typical still image


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                                                                                                       ISSN 1947-5500
                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                         Vol. 9, No. 1, January 2011
                                                                                                                                             2

        The basis vectors of the Haar matrix are sequency                        IV. IMPLEMENTATION METHODOLOGY
         ordered.
        Haar Transform has poor energy compaction for                       Each image [27] is presented mathematically by a matrix
         images.                                                          of numbers. Haar wavelet uses a method for manipulating the
        Orthogonality: The original signal is split into a low           matrices called averaging and differencing. Entire row of a
         and a high frequency part, and filters enabling the              image matrix is taken, then do the averaging and differencing
         splitting without duplicating information are said to be         process. After we treated entire each row of an image matrix,
         orthogonal.                                                      then do the averaging and differencing process for the entire
        Linear Phase: To obtain linear phase, symmetric filters          each column of the image matrix. Then consider this matrix
         would have to be used.                                           is known an semifinal matrix (T) whose rows and columns
        Compact support: The magnitude response of the filter            have been treated.       This procedure is called wavelet
         should be exactly zero outside the frequency range               transform.
         covered by the transform. If this property is satisfied,                  Then compare the original matrix and last matrix
         the transform is energy invariant.                               that is semifinal matrix(T), the data has became smaller.
        Perfect reconstruction: If the input signal is                   Since the data has become smaller, it is easy to transform and
         transformed and inversely transformed using a set of             store the information. The important one is that the treated
         weighted basis functions, and the reproduced sample              information is reversible. To explain the reversing process
         values are identical to those of the input signal, the           we need linear algebra. Using linear algebra is to maximize
         transform is said to have the perfect reconstruction             compression while maintaining a suitable level of detail.
         property. If, in addition no information redundancy is
         present in the sampled signal, the wavelet transform is,         A. Wavelet Compression Methodology
         as stated above, ortho normal.
                                                                             From Semi final Matrix (T) is ready to be compressed. [29]
  No wavelets can possess all these properties, so the choice             Definition of Wavelet Compression is fix a non negative
of the wavelet is decided based on the consideration of which             threshold value ε and decree that any detail coefficient in the
of the above points are important for a particular application.           wavelet transformed data whose magnitude is less than or
Haar-wavelet, Daubechies-wavelets and biorthogonal-                       equal to zero (this leads to a relatively sparse matrix). Then
wavelets are popular choices [1]. These wavelets have                     rebuild an approximation of the original data using this
properties which cover the requirements for a range of                    doctored version of the wavelet transformed data. In the case
applications.                                                             of image data, we can throw out a sizable proportion of the
                                                                          detail coefficients in this and obtain visually acceptable
The advantages of Haar Wavelet transform as follows:                      results. This process is called lossless compression. When no
                                                                          information is loss (eg., if =0). Otherwise it is referred to as
1.   Best performance in terms of computation time.                       lossy compression (in which case  >0). In the former case,
2.   Computation speed is high.                                           we can get our original data back, and in the latter we can
3.   Simplicity                                                           build an approximation of it. Because we know this, we can
4.   HWT is efficient compression method.                                 eliminate some information from our matrix and still be
5.   It is memory efficient, since it can be calculated inplace           capable of attaining a fairly good approximation of our
     without a temporary array.                                           original matrix. Doing this, we take threshold value ε=10
                                                                          ie., reset to zero all elements of semifinal matrix(T) which are
                                                                          less than or equal to 10 in absolute value. From this we obtain
         III. PROCEDURE FOR HAAR WAVELET TRANSFORM                        the doctored matrix. Then apply the inverse wavelet
                                                                          transform to doctored matrix we get the reconstructed
To calculate the Haar transform of an array of n samples:                 approximation R.
                                                                                     In this process, we can get a good approximation of
1. Find the average of each pair of samples. (n/2                         the original image. We have lost some of the detail in the
   averages)                                                              image but it is so minimal that the loss would not be
2. Find the difference between each average and the samples               noticeable in most cases.
   it was calculated from. (n/2 differences)
3. Fill the first half of the array with averages.
4. Fill the second half of the array with differences.
5. Repeat the process on the first half of the array. (The array
   length should be a power of two)


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                                                                                                      ISSN 1947-5500
                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                       Vol. 9, No. 1, January 2011
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B. Linear Algebra Methodology                                                    Display the resulting images and comment on the
                                                                                  quality of the images.
   To apply the averaging and differencing using linear
                                                                                 Calculate MSE, MAE & PSNR values of different
algebra [27] we can use matrices such as A1,A2,A3…..An.
                                                                                  Compression Ratios for corresponding Reconstructed
that perform each of the steps of the averaging and                               images.
differencing process.
                                                                                 Then add a small amount of white noise to the input
                                                                                  image. Default : variance =0.01, sigma=0.1, mean=0
 i. When multiplying the string by the first matrix of the
    first half of columns are taking the average of each pair                    To compute the haar wavelet transform, set all the
    and the last half of columns take the corresponding                           approximation coefficients to zero except those
    differences.                                                                  whose magnitude is larger than 3 sigma.

ii. The second matrix works in much the same way, the first                      This same case is applicable to detail coefficients
    half of columns now perform the averaging and                                 that is horizontal, vertical & diagonal coefficients.
    differencing to the remaining pairs, and the identity                        Reconstruct an estimate of the original image by
    matrix in the last half of columns carry down the detail                      applying the corresponding inverse transform.
    coefficient from step i.
                                                                                 Display and compare the results by computing the
iii. Similarly in the final step, the averaging and differencing                  root mean square error, PSNR, and mean absolute
     is done by the first two columns of the matrix, and the                      error of the noisy image and the denoising image.
     identity matrix carries down the detail coefficient from
                                                                                 The same process is repeated for various images and
     previous step.
                                                                                  compare its performance.
iv. To simplify this process, we can multiply these matrices
    together to obtain a single transform matrix                         Alternative approach Algorithm is described as follows:
    W=A1A2A3) we can now multiply our original string by
    just one transform matrix to go directly form the original           1.   Read the image cameraman.tif from the user.
    string to the final results of step iii.
                                                                         2.   Using 2D wavelet decomposition with respect to a haar
v. In the following equation we simplify this process of                      wavelet computes the approximation coefficients matrix
   matrix multiplication. First the averaging and                             CA and detail coefficient matrixes CH, CV, CD
   differencing and second the inverse of those operation.                    (horizontal, vertical & diagonal respectively) which is
                                                                              obtained by wavelet decomposition of the input matrix
                                                                              ie., im_input.
                   1. T= ((AW)T W))T
                      T= (WT AT W)T                                      3.   From this, again using 2D wavelet decomposition with
                                                                              respect to a haar wavelet computes the approximation
                      T= WT (AT)T (WT)T
                                                                              and detail coefficients which are obtained by wavelet
                      T=WTAW                                (3)
                                                                              decomposition of the CA matrix. This is considered as
                                                                              level 2.
                   2. (W1)-1 T W-1 = A
                      (W-1)1TW-1=A                          (4)          4.   Again apply the haar wavelet transform from CA matrix
                                                                              which is considered as CA1 for level 3.
                                                                         5.   Do the same process and considered as CA2 for level 4
                      V. ALGORITHM                                       6.   Take inverse transform for level 1, level 2, level 3 &
                                                                              level 4 that ie., im_input, CA, CA1, CA2.
        Read the image from the user.
                                                                         7.   Reconstruct the images for level 1, level 2, level 3 &
        Apply 2 D DWT using haar wavelet over the image                      level 4.
        For the computation of haar wavelet transform, set              8.   Display the results of reconstruction 1, reconstruction 2,
         the threshold value 25%, 10%, 5%, 1% ie., set all                    reconstruction 3, reconstruction 4 ie., level 1 , 2 , 3 , 4
         the coefficients to zero except for the largest in                   with respect to the original image.
         magnitude 25%, 10%, 5%, 1% . And reconstruct an
         approximation to the original image by apply the
         corresponding inverse transform with only modified
         approximation coefficients.
        This simulates the process of compressing by factors
         of ¼,1/10,1/20,1/100 respectively.

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                                                                                                     ISSN 1947-5500
                                                           (IJCSIS) International Journal of Computer Science and Information Security,
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            VI. METRICS FOR COMPARISON                                                                                              
                                                                                                                     2              
                                                                                PSNR=10xlog10                    255                                          (9)
A. Compression Metrics                                                                          ! N1 N 2                          2 
                                                                                                        ( f ( x, y)  g ( x, y)) 
The type of compression metrics [21] used for digital data                                      N1 N 2 y 1 z 1                    
varies quite markedly depending on the type of compression                  And also Mean Absolute Error is denoted by
employed that use a simple ratio formula as given below
                                                                                                                     
                                                                                                                 1        N
                           Output File size (bytes)                                            MAE=                              |g(x,y)-f(x,y)|               (10)
   CompressionRatio(%)=                             x100 (5)                                                     N        i 1
                            Input File size (bytes)
                                                                              The standard error measures a large distortion, but the
  A measure of rate is far more suitable metric for large
                                                                            image has merely been brightened. Individually, MSE, SNR
compression ratios as it gives the number of bits per pixel
                                                                            and PSNR are not very good at measuring subjective image
used to encode an image rather than some abstract
                                                                            quality, but used together these error metrics are at least
percentage.
                                                                            adequate at determining if an image is reproduced at a certain
                   8 x Output File size (bytes)                             quality.
        Rate(bpp)=                                      (6)
                      Input File size (bytes)
   Rate, for image coding purposes, uses bits per pixel (bpp)                                VII. RESULTS AND DISCUSSIONS
as its unit.
                                                                                 The project deals with the implementation of the haar
B. Error Metrics                                                            wavelet compression techniques and a comparison over
                                                                            various input images. We first look in to results of wavelet
   In general, error measurements are used on lossy                         compression technique by calculating their comparison ratios
compressed images to try and quantify the quality of a                      and then compare their results based on the error metrics
picture. Getting a quantifiable measure of the distortion                   which is shown in Table I.
between two images is very important as one can try and                                                 Table I
minimize this thesis so as to better replicate the original                    Different types of Error Metrics with respect to Various
image. There are many ways of measuring the fidelity of a                                        Compression Ratios.
picture g(x,y) to its original f(x,y). One of the simplest and                 Compression                      Error Metrics
most popular methods is to use the difference between f and                       Ratios             MSE            RMSE          PSNR
g. In its most basic form is the mean square error (MSE) [11]                         4:01         9937.92         99.6891       18.7841
given by,                                                                             10:01        14380.1         119.917       15.0893
              1      N1     N2                                                        20:01        15990.1         126.452        14.028
    MSE=
            N1 N 2
                       ( f ( x, y)  g ( x, y))
                     y 1   z 1
                                                    2
                                                                (7)                   100:1        17453.28       132.1109       13.1524

where f and g are N1xN2 size image. This is a very useful                   A. Effects on Compression Ratio Vs Various Parameters
measure as it gives an average value of the energy loss in the
lossy compression of the original image f. Signal-to-noise                  Wavelet Compression is applied for all images and the
ratio (SNR) is another measure often used to compare the                    compression ratio is being calculated. The image quality thus
performance of reproduced images which is defined by,                       measured in compression techniques is compared using a
                                                                            BAR CHART, which proves the image quality of the various
                            ! N1 N 2                                      input images which reconstructed images are better. This is
                 
    SNR=10xlog10 
                                    f ( x, y) 2
                         N1 N 2 y 1 z 1
                                                       
                                                                (8)         shown in Fig 1 & 2.
                                                                                                   COMPRESSION RATIO VS MSE
                  ! N1 N 2                          2 
                          ( f ( x, y)  g ( x, y)) 
                  N1 N 2 y 1 z 1                                                 50000
                                                                                     40000
                                                                                                                                                  4:01
                                                                                     30000
                                                                               MSE




                                                                                                                                                  10:01
SNR is measured in dB's and gives a good indication of the                           20000
                                                                                                                                                  20:01
ratio of signal to noise reproduction. This is a very similar                        10000                                                        100:01:00
measurement to MSE. A more subjective qualitative                                        0
measurement of distortion is the Peak Signal-to-noise ratio
                                                                                                                                if
                                                                                                                               if




                                                                                                                                         tif
                                                                                                             f
                                                                                                f


                                                                                                         .ti
                                                                                             .ti




                                                                                                                           r.t
                                                                                                                            t.t




                                                                                                                                           .
                                                                                                                                        ce
                                                                                                      ri
                                                                                          an




                                                                                                                         ar
                                                                                                                   gh




(PSNR) [3].
                                                                                                     m




                                                                                                                                     ri
                                                                                        cm




                                                                                                                        m
                                                                                                                 ei


                                                                                                                     ne
                                                                                                                        bo




                                                                                                                     IMAGES
                                                                              Fig. 1. Effects on Compression Ratio Vs MSE with Respect to Various
                                                                              Input Images


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                                                                                          PSNR values depend on image type and cannot be used to
                            COMPRESSION RATIO Vs PSNR                                     compare images with different content.
        70

        60
                                                                        4:01
                                                                                            The quality of the reconstructed image is measured using
        50                                                                                the error metrics:
                                                                        10:01
        40
                                                                                                              MSE
 PSNR




        30                                                              20:01
                                                                                                              PSNR
        20                                                              100:01:00
        10
                                                                                          The values of MSE & PSNR are calculated for all test images
        0
                                                                                          and are provided in Table II.
             cman.tif   mri.tif   eight.tif   bonemarr.tif   rice.tif

                                  IMAGES
                                                                                                                      Table II
         Fig. 2. Effects on Compression Ratio Vs PSNR with                                Calculation of different kinds of Error Metrics with respect to
                    Respect to various input images                                          Various Input images under Compression Ratio 100:1

  In the first experiment, we report on Compressed Ratio                                                            No of
consumed by Error Metrics as described in the previous                                    SI.No        Image     elements to              MSE           PSNR
section. In our experiments, we used the gray scale sample,                                                     store in bytes
cameraman.tif of size 256*256. And measured the                                             1        cman.tif     256*256              17543.280       13.1524
compression ratio and the PSNR of the compressed image. In                                  2        Rice.tif     256*256              13094.653       16.0257
each case, the threshold level was changed and got the output.                              3         Mri.tif     128*128               1248.827       39.5257
The results are presented in Fig 5. The x-axis represents                                   4        Eight.tif    242*308              41227.182       4.5567
Compression Ratio, while the y-axes represent MSE with                                      5      Bonemarr.tif   238*270              28009.786       8.4222
various input images.

   In the Second experiment, we report on Compression Ratio                                                    VIII. CONCLUSION
Vs PSNR with various input images. In our experiments, we
used the gray scale image samples. In each case, the threshold                               This paper reported is aimed at developing computationally
level was changed and got the output. The results are                                     efficient and effective algorithm for lossy image compression
presented in Fig 5. The x-axis represents Compression ratio,                              using wavelet techniques. This paper is particularly targeted
while the y-axes represents PSNR with various input                                       towards wavelet image compression using Haar
images.                                                                                   Transformation with an idea to minimize the computational
                                                                                          requirements to achieve good reproduction image quality. In
   The quality of compressed image depends on the no of                                   this direction the following methods are developed. This
decompositions. The no of decompositions determines the                                   image compression schemes for images have been presented
resolution of the lowest level in wavelet domain. Using larger                            based on the 2 D HWT. The promising results obtained
no of decompositions, that will be more successful in                                     concerning reconstructed image quality as well as
resolving important HWT coefficients from less important                                  preservation of significant image details, while on the
coefficients. After decomposing the image and representing                                otherhand achieving high compression rates.
it with wavelet coefficients, compression can be performed by
ignoring all coefficients below some threshold. In this                                      High compression ratio and better image quality
experiment, compression is obtained by wavelet coefficient                                    obtained which is better than existing methods.
thresholding. All coefficients below some threshold are
                                                                                             In addition, the above methods are to be for noisy
neglected and Compression Ratio is computed. Compression
                                                                                              images
Algorithm operation is follows : Compression Ratio is fixed
to the required level and threshold value has been changed to                                To improve the quality of the reconstructed image
achieve required Compression Ratio after that PSNR is                                        The results are executed in fraction of seconds.
computed. PSNR tends to saturate for a larger no of
                                                                                             To obtain the wavelet coefficients are nearly closer to
decompositions. For each compression ratio, the PSNR
                                                                                              zero.
characteristic has “threshold” which represents the optimal
no of decompositions. Below and above the threshold PSNR                                     Image denoising techniques will allow precise imaging
decreases and no of decomposition increases. PSNR is                                          at much faster rates by greatly reducing the necessary
increased up to some no of decompositions. Beyond that,                                       averaging time to construct low noise images.
increasing the no of decomposition has a negative effect.


                                                                                    131                              http://sites.google.com/site/ijcsis/
                                                                                                                     ISSN 1947-5500
                                                      (IJCSIS) International Journal of Computer Science and Information Security,
                                                      Vol. 9, No. 1, January 2011
                                                                                                                                                   6

   The performance of an image compression algorithm is                 implementation issues such as bit allocation methods and
basically evaluated in terms of compression ratio, MSE, and             error estimation can be studied.
PSNR. A `good' algorithm has a high compression ratio.
Wavelet based image compression theory has rapidly                        Image denoising method using wavelet for noisy image
increased in the last seven years. With the increasing use of           could be developed. This yield better result in image
multimedia technologies image compression requires higher               compression techniques using wavelet for noisy input images.
performance as well as new functionality. To address this
need in the specific area of still image compression here a                                          IX. REFERENCES
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                               Fig. 3: Original and Haar Wavelet Transformed images for Different Levels




                                  Fig. 4: Reconstructed Images for different Levels



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